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A070932 Possible number of units in a finite (commutative or non-commutative) ring. 3
0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 36, 40, 42, 44, 45, 46, 48, 49, 52, 54, 56, 58, 60, 62, 63, 64, 66, 70, 72, 78, 80, 81, 82, 84, 88, 90, 92, 93, 96, 98, 100 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This is a list of the numbers of units in R where R ranges over all finite commutative or non-commutative rings.

By considering the ring Z_n and the finite fields GF(q) this sequence contains the values of the Euler function phi(n) (A000010) and prime powers - 1 (A181062). By taking direct product of rings, if n and m belong to the sequence then so does m*n.

Eric M. Rains has shown that these rules generate all terms of this sequence. More precisely, he shows this sequence (with 0 removed) is the multiplicative monoid generated by all numbers of the form q^n-q^{n-1} for n >= 1 and q a prime power (see Rains link).

Since the number of units of F_q[X]/(X^n) is q^n - q^(n-1), restricting to finite commutative rings gives the same sequence. A296241, which is a proper supersequence, allows the ring R to be infinite. - Jianing Song, Dec 24 2021

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

E. M. Rains, Comments on A070932

MATHEMATICA

max = 100; A000010 = EulerPhi[ Range[2*max]] // Union // Select[#, # <= max &] &; A181062 = Select[ Range[max], Length[ FactorInteger[#]] == 1 &] - 1; FixedPoint[ Select[ Outer[ Times, #, # ] // Flatten // Union, # <= max &] &, Union[A000010, A181062] ] (* Jean-Fran├žois Alcover, Sep 10 2013 *)

PROG

(PARI) list(lim)=my(P=1, q, v, u=List()); forprime(p=2, default(primelimit), if(eulerphi(P*=p)>=lim, q=p; break)); v=vecsort(vector(P/q*lim\eulerphi(P/q), k, eulerphi(k)), , 8); v=select(n->n<=lim, v); forprime(p=2, sqrtint(lim\1+1), P=p; while((P*=p) <= lim+1, listput(u, P-1))); v=vecsort(concat(v, Vec(u)), , 8); u=List([0]); while(#u, v=vecsort(concat(v, Vec(u)), , 8); u=List(); for(i=3, #v, for(j=i, #v, P=v[i]*v[j]; if(P>lim, break); if(!vecsearch(v, P), listput(u, P))))); v \\ Charles R Greathouse IV, Jan 08 2013

CROSSREFS

Cf. A000010, A002202, A000252, A000961, A181062, A221178.

A000252 is a subsequence.

A282572 is the subsequence of odd terms.

Proper subsequence of A296241.

The main entries concerned with the enumeration of rings are A027623, A037234, A037291, A037289, A038538, A186116.

Sequence in context: A238369 A296858 A296241 * A161577 A093686 A325031

Adjacent sequences:  A070929 A070930 A070931 * A070933 A070934 A070935

KEYWORD

nonn,nice

AUTHOR

Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002

EXTENSIONS

Entry revised by N. J. A. Sloane, Jan 06 2013, Jan 08 2013

Definition clarified by Jianing Song, Dec 24 2021

STATUS

approved

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Last modified October 7 08:05 EDT 2022. Contains 357270 sequences. (Running on oeis4.)